Question

Find the value of the expression :
${\tan }^{-1}\left(\tan \frac{3\pi }{4}\right)$

Answer 1

Given :
${\tan }^{-1}\left(\tan \frac{3\pi }{4}\right)$
We know that the range of the principal value branch of ${\tan }^{-1}$ is $(-\frac{\pi }{2},\frac{\pi }{2})$.
$\therefore {\tan }^{-1}\left(\tan (\pi -\frac{\pi }{4})\right)$
$={\tan }^{-1}(-\tan \frac{\pi }{4})=-\frac{\pi }{4}$
Hence, the value of the expression is $-\frac{\pi }{4}.$